Problem 56 - Calculus Volume 1

Question

An American tourist visits Paris and must convert U.S. dollars to Euros, which can be done using the function \(E(x)=0.79 x,\) where \(x\) is the number of U.S. dollars and \(E(x)\) is the equivalent number of Euros. Since conversion rates fluctuate, when the tourist returns to the United States 2 weeks later, the conversion from Euros to U.S. dollars is \(D(x)=1.245 x,\) where \(x\) is the number of Euros and \(D(x)\) is the equivalent number of U.S. dollars. a. Find the composite function that converts directly from U.S. dollars to U.S. dollars via Euros. Did this tourist lose value in the conversion process? b. Use (a) to determine how many U.S. dollars the tourist would get back at the end of her trip if she converted an extra \(\$ 200\) when she arrived in Paris.

Step-by-Step Solution

Answer

The tourist loses value in the conversion. After starting with $200, she returns with $196.71.

Step 1: Define Composite Function for Conversion

To find the composite function that converts U.S. dollars to U.S. dollars via Euros, we need to combine the functions \(E(x)\) and \(D(x)\). This involves finding \(D(E(x))\). The function \(E(x) = 0.79x\) converts U.S. dollars to Euros, and \(D(x) = 1.245x\) converts Euros back to U.S. dollars.

Step 2: Derive Composite Function \(D(E(x))\)

The composite function \(D(E(x))\) is found by substituting \(E(x)\) into \(D(x)\): \[ D(E(x)) = D(0.79x) = 1.245(0.79x) = (1.245 \times 0.79)x = 0.98355x. \] This shows that each U.S. dollar initially converted will return as \(0.98355\) dollars after conversion to Euros and back to dollars.

Step 3: Analyze Value Loss in Conversion Process

The composite function \(D(E(x)) = 0.98355x\) indicates that after converting money to Euros and back, each U.S. dollar is worth \(0.98355\) dollars. Since \(0.98355 < 1\), the tourist indeed loses money in the conversion process.

Step 4: Calculate Return for Extra \$200

To find out how much the tourist gets back for \\(200 initially converted, substitute \(x = 200\) into the composite function: \[ D(E(200)) = 0.98355 \times 200 = 196.71. \] Thus, the tourist would get \\)196.71 back after converting the extra \$200.

Key Concepts

Currency Conversion

Currency conversion is the process of exchanging one nation's currency for another. This is crucial for travelers, businesses, and investors who engage internationally. Currency exchange rates dictate how much one will receive of another currency.
These rates fluctuate constantly due to factors like economic stability, inflation, and trade dynamics. Understanding currency conversion rates is essential for making informed financial decisions abroad.

• **Exchange Rate**: It is the rate at which one currency can be exchanged for another. For example, the conversion from U.S. dollars (USD) to Euros (EUR) in this exercise uses the function \(E(x)=0.79x\), meaning 1 USD equals 0.79 EUR.
• **Regular Monitoring**: Always check the current rates, as fluctuations can significantly impact the value of your money.

This basic understanding should guide anyone involved in international transactions.

Function Composition

In mathematics, a composite function is created when one function is applied and then another function is applied to the result of the first. In our currency conversion problem, we use this concept to convert U.S. dollars to Euros and back to U.S. dollars.

To form a composite function, you combine two functions, say \(E(x)\) and \(D(x)\). Here, \(E(x) = 0.79x\) converts dollars to euros, and \(D(x) = 1.245x\) brings it back. We seek the composite function \(D(E(x))\) which directly shows the conversion back and forth:- **Define the Function**: Substitute the Euro conversion function into the dollar conversion: \[ D(E(x)) = 1.245(0.79x) \ = 0.98355x. \]- **Utility**: This function indicates that starting with \( x \) U.S. dollars, you end up with only \( 0.98355x \) after converting twice. This is a critical realization in understanding how multiple conversions affect the value. It exemplifies function composition where one operation follows the next seamlessly.

Value Loss in Currency Exchange

When converting currency back and forth between two countries, there can often be a loss in value. This is evident once the composite function \(D(E(x))\) is determined. In this case, the composite function shows that using this specific conversion process, each U.S. dollar returns as \(0.98355\) dollars.
This means that a tourist loses money when converting first to Euros and then back to Dollars.

• **Understanding Value Loss**: The derived function, \(D(E(x)) = 0.98355x\), tells us that for every dollar converted, it returns slightly less.
• **Impact Calculations**: For example, converting an additional \(200 results in receiving only \)196.71 back. Functions like these highlight hidden costs in financial processes.

Comprehending these losses is crucial for effective financial planning, especially for those frequently engaging in international travel or business.